## Notre Dame Journal of Formal Logic

### Club Guessing and the Universal Models

Mirna ǅamonja

#### Abstract

We survey the use of club guessing and other PCF constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal, element.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 46, Number 3 (2005), 283-300.

Dates
First available in Project Euclid: 30 August 2005

https://projecteuclid.org/euclid.ndjfl/1125409327

Digital Object Identifier
doi:10.1305/ndjfl/1125409327

Mathematical Reviews number (MathSciNet)
MR2160658

Zentralblatt MATH identifier
1105.03037

Keywords
universal models club guessing

#### Citation

ǅamonja, Mirna. Club Guessing and the Universal Models. Notre Dame J. Formal Logic 46 (2005), no. 3, 283--300. doi:10.1305/ndjfl/1125409327. https://projecteuclid.org/euclid.ndjfl/1125409327

#### References

• [1] Argyros, S. A., and Y. Benyamini, "Universal WCG Banach spaces and universal Eberlein compacts", Israel Journal of Mathematics, vol. 58 (1987), pp. 305--20. špace-1.5pt
• [2] Bell, M., "Universal uniform Eberlein compact spaces", Proceedings of the American Mathematical Society, vol. 128 (2000), pp. 2191--97. špace-1.5pt
• [3] Benyamini, Y., M. E. Rudin, and M. Wage, "Continuous images of weakly compact subsets of Banach spaces", Pacific Journal of Mathematics, vol. 70 (1977), pp. 309--24. špace-1.5pt
• [4] Chang, C. C., and H. J. Keisler, Model Theory, 2d edition, vol. 73 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1977. špace-1.5pt
• [5] Cherlin, G. L., "The classification of countable homogeneous directed graphs and countable homogeneous $n$-tournaments", Memoirs of the American Mathematical Society, vol. 131, no. 621 (1998). špace-1.5pt
• [6] Dow, A., "Remarks on recent results in set-theoretic topology", preprint, 2005.špace-1.5pt
• [7] Džamonja, M., "Universality of uniform Eberlein compacta", forthcoming in the Proceedings of the American Mathematical Society.špace-1.5pt
• [8] Džamonja, M., "On uniform Eberlein compacta and c-algebras", Topology Proceedings, vol. 23 (1998), pp. 143--50. špace-1.5pt
• [9] Džamonja, M., and S. Shelah, "On properties of theories which preclude the existence of universal models", forthcoming in the Annals of Pure and Applied Logic.špace-1.5pt
• [10] Džamonja, M., and S. Shelah, "On the existence of universal models", Archive for Mathematical Logic, vol. 43 (2004), pp. 901--36. špace-1.5pt
• [11] Džamonja, M., and K. Thompson, "Universality results for well-founded posets", submitted to Sarajevo Journal of Mathematics.špace-1.5pt
• [12] Foreman, M., and P. Komjáth, "The club guessing ideal (commentary on a theorem of Gitik and Shelah)", preprint, 2003.špace-1.5pt
• [13] Füredi, Z., and P. Komjáth, "Nonexistence of universal graphs without some trees", Combinatorica, vol. 17 (1997), pp. 163--71. špace-1.5pt
• [14] Ishiu, T., Club Guessing Sequences and Filters, Ph.D. thesis, University of California, Irvine, 2003.špace-1pt
• [15] Kojman, M., and S. Shelah, "Almost isometric embeddings between metric spaces", preprint, 2003.špace-1.5pt
• [16] Kojman, M., "Representing embeddability as set inclusion", Journal of the London Mathematical Society. Second Series, vol. 58 (1998), pp. 257--70. špace-1.5pt
• [17] Kojman, M., and S. Shelah, "Nonexistence of universal orders in many cardinals", The Journal of Symbolic Logic, vol. 57 (1992), pp. 875--91. špace-1.5pt
• [18] Kojman, M., and S. Shelah, "The universality spectrum of stable unsuperstable theories", Annals of Pure and Applied Logic, vol. 58 (1992), pp. 57--72. špace-1pt
• [19] Kojman, M., and S. Shelah, "Universal Abelian groups", Israel Journal of Mathematics, vol. 92 (1995), pp. 113--24. špace-1pt
• [20] Komjáth, P., and S. Shelah, "Universal graphs without large cliques", Journal of Combinatorial Theory. Series B, vol. 63 (1995), pp. 125--35. špace-1pt
• [21] Mekler, A., and J. Väänänen, "Trees and $\Pi\sp 1\sb 1$"-subsets of $\sp \omega\sb 1\omega\sb 1$, The Journal of Symbolic Logic, vol. 58 (1993), pp. 1052--70. špace-1pt
• [22] Rado, R., "Universal graphs and universal functions", Acta Arithmetica, vol. 9 (1964), pp. 331--40. špace-1pt
• [23] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, 2d edition, vol. 92 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990. špace-1pt
• [24] Shelah, S., "The universality spectrum: Consistency for more classes", pp. 403--20 in Combinatorics, Paul Erdös is Eighty, Vol. 1, Bolyai Society Mathematical Studies, János Bolyai Mathematical Society, Budapest, 1993. Proceedings of the Meeting in honor of P. Erdös, Keszthely, Hungary, 1993. Improved version available at \hrefhttp://www.math.rutgers.edu/~shelarchhttp://www.math.rutgers.edu/\~ shelarch. špace-1pt
• [25] Shelah, S., Analytical guide and corrections to Cardinal Arithmetic,'' preprint at \hrefhttp://www.math.rutgers.edu/~shelarchhttp://www.math.rutgers.edu/\~ shelarch, \hrefhttp://front.math.ucdavis.edu/math.LO/9906022http://front.math.ucdavis.edu/math.LO/9906022.špace-1pt
• [26] Shelah, S., "Existence of universal structures", in preparation.špace-1pt
• [27] Shelah, S., "Notes on Abelian groups", in preparation.špace-1pt
• [28] Shelah, S., "Independence results", The Journal of Symbolic Logic, vol. 45 (1980), pp. 563--73. špace-1pt
• [29] Shelah, S., "On universal graphs without instances of CH", Annals of Pure and Applied Logic, vol. 26 (1984), pp. 75--87. špace-1pt
• [30] Shelah, S., "Universal graphs without instances of CH: Revisited", Israel Journal of Mathematics, vol. 70 (1990), pp. 69--81. špace-1pt
• [31] Shelah, S., "Advances in cardinal arithmetic", pp. 355--83 in Finite and Infinite Combinatorics in Sets and Logic (Banff AB, 1991), vol. 411 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Kluwer Academic Publishers, Dordrecht, 1993. špace-1pt
• [32] Shelah, S., Cardinal Arithmetic, vol. 29 of Oxford Logic Guides, The Clarendon Press, New York, 1994. špace-1pt
• [33] Shelah, S., "Toward classifying unstable theories", Annals of Pure and Applied Logic, vol. 80 (1996), pp. 229--55. špace-1pt
• [34] Shelah, S., "Universal in $(<\lambda)$-stable Abelian group", Mathematica Japonica, vol. 44 (1996), pp. 1--9. špace-1pt
• [35] Shelah, S., "Non-existence of universals for classes like reduced torsion free Abelian groups under embeddings which are not necessarily pure", pp. 229--86 in Advances in Algebra and Model Theory (Essen, 1994; Dresden, 1995), vol. 9 of Algebra, Logic and Applications, Gordon and Breach, Amsterdam, 1997. špace-1pt
• [36] Shelah, S., Proper and Improper Forcing, 2d edition, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1998. špace-1pt
• [37] Shelah, S., "On what I do not understand (and have something to say), Model Theory", Mathematica Japonica, vol. 51 (2000), pp. 329--77. špace-1pt
• [38] Shelah, S., "Non-existence of universal members in classes of Abelian groups", Journal of Group Theory, vol. 4 (2001), pp. 169--91. špace-1pt
• [39] Shelah, S., Is there universality among the graph omitting a complete bipartite graph?'' preprint, 2004.špace-1pt
• [40] Shelah, S., and A. Usvyatsov, "Model theoretic stability for complete metric spaces", in preparation.špace-1pt
• [41] Shelah, S., and A. Usvyatsov, "Banach spaces and groups---order properties and universal models", preprint, 2002.špace-1pt
• [42] Szlenk, W., "The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces", Studia Mathematica, vol. 30 (1968), pp. 53--61. špace-1pt
• [43] Thompson, K., Universality results for ordered and directed structures, Ph.D. thesis, University of East Anglia, 2003.špace-1pt
• [44] Thompson, K., "Universality for orders and graphs which omit large substructures", submitted to Notre Dame Journal of Formal Logic.špace-1pt
• [45] Todorčević, S., "The functor $\sigma\sp 2 x$", Studia Mathematica, vol. 116 (1995), pp. 49--57. špace-1pt
• [46] Usvyatsov, A., Ph.D. thesis, Hebrew University of Jerusalem, in preparation.